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Tan, Cos, Sin


Date: 05/21/99 at 17:30:09
From: Siobhan
Subject: tan, cos, sin

Please tell me about tan, sin, and cos. Also, is there a way of 
working them out without a calculator? How do you know which one to 
use?


Date: 05/22/99 at 16:19:49
From: Doctor Jeremiah
Subject: Re: tan, cos, sin

Hi there.

Trigonometry is complicated until you understand it. Then you wonder 
why it seemed so complicated.

First you need a triangle, but not just any triangle. You need a 
triangle that has one angle of 90 degrees.

                +
               /|
              / |
             /  |
            /   |
           /    |
          /     |
         /      |
        c       b
       /        |
      /         |
     /          |
    / x       90|
   +------a-----+

I have labelled the sides a, b and c. I also labelled one angle as x.

The sine of the angle x is written as sin(x).
The cosine of the angle x is written as cos(x).
The tangent of the angle x is written as tan(x).

Each of these values is just a fraction made of the lengths of two of 
the triangle's sides.

For example:

sin(x) is a fraction made by taking the length of the short side 
opposite x and dividing it by the long side.
Written as an equation    sin(x) = b/c

cos(x) is a fraction made by taking the length of the short side next 
to x and dividing it by the long side.
Written as an equation    cos(x) = a/c

tan(x) is a fraction made by taking the length of the short side 
opposite x and dividing it by the length of the short side next to x.
Written as an equation    tan(x) = b/a


Let's say that we know the lengths of two sides of the triangle below:

                +
               /|
              / |
             /  |
            /   |
           /    |
          /     |
         /      |
        /       | 12
       /        |
      /         |
     /          |
    / x       90|
   +------------+
          5

We can find the tangent of the angle x. This we can do because tan(x) 
is a fraction made by taking the length of the short side opposite x 
(12) and dividing it by the length of the short side next to x (5).

tan(x) = 12/5 = 2.4

Let's say that we know only one side of the triangle but that we know 
the angle as well:

                +
               /|
              / |
             /  |
            /   |
           /    |
          /     |
         /      |
        c       | 12
       /        |
      /         |
     /          |
    / 30      90|
   +------a-----+
          
Here we can find the lengths of either side.

- If we use the equation for the sine of x, then when we put the angle 
and the opposite side's length in we get sin(30) = 12/c, and after
rearranging we would get c = 12/sin(30).

- If we use the equation for tangent, then when we put the angle and 
the opposite side in we get tan(30) = 12/a, and after rearranging
we would get a = 12/tan(30).

The problem here is that we don't know the value of the sine of 30 
degrees. Can we figure it out without a calculator? Sure. There are 
several ways, but they are complicated to explain. It turns out that 
the sine of 30 is 0.5.

One method of calculating sines and cosines is to use a Taylor series.  
The tangent of an angle is calculated differently. The tangent is just
the sine divided by the cosine.

Let me know if you need more help.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Trigonometry

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